Show that,$$\int_0^\pi \bigg|\dfrac{\sin nx}{x}\bigg|\mathrm{d}x \ge \dfrac{2}{\pi}\bigg(1+\dfrac12+\cdots+\dfrac{1}{n}\bigg)$$

I could not approach the problem at all. Please help.

  • 4
    $\begingroup$ The zeros of $\sin nx$, where it changes sign, are $\dfrac{k\pi}{n},\; k = 0,\,1,\,\dotsc,\,n$. Splitting the integral and looking at each part alone ought to help. $\endgroup$ – Daniel Fischer Feb 20 '14 at 14:35
  • $\begingroup$ @DanielFischer Let me try...thank you for the hint... $\endgroup$ – Hawk Feb 20 '14 at 14:35
  • $\begingroup$ @DanielFischer I got the representation as you said...but, would I need to evaluate each and every integral or I would need to get another function which looks similar to the above function but is less than or equal to the above expression to get the inequality? $\endgroup$ – Hawk Feb 20 '14 at 14:44
  • $\begingroup$ See math.stackexchange.com/a/390841/4583 $\endgroup$ – Ayman Hourieh Feb 20 '14 at 14:56
  • $\begingroup$ @AymanHourieh I would really appreciate it a lot if you could post a solution here without using the summation notion. I face very much difficulty to conceive the representation with summation. $\endgroup$ – Hawk Feb 20 '14 at 15:02

$$ \begin{align} I &=\sum_{i=0}^{n-1} \int_{i\pi/n}^{(i+1)\pi/n} \frac{|\sin{nx}|}{|x|}\,dx \\ &=\sum_{i=0}^{n-1} \int_{0}^{\pi/n} \frac{|\sin{nx}|}{i\pi/n+ x}\,dx \\ &\gt \sum_{i=1}^{n} \int_{0}^{\pi/n} \frac{|\sin{nx}|}{i\pi/n}\,dx \\ &= \sum_{i=1}^{n} \frac{2/n}{i\pi/n} \\ &= \frac{2}{\pi}\sum_{i=1}^{n} \frac{1}{i} \end{align} $$


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